Some geometric characterizations of extremal sets in hilbert spaces

HNUE JOURNAL OF SCIENCE DOI 10 18173/2354 1059 2022 0019 Natural Science, 2022, Volume 67, Issue 2, pp 19 24 This paper is available online at http //stdb hnue edu vn SOME GEOMETRIC CHARACTERIZATIONS[.]

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2022-0019 Natural Science, 2022, Volume 67, Issue 2, pp 19-24

This paper is available online at http://stdb.hnue.edu.vn

SOME GEOMETRIC CHARACTERIZATIONS OF EXTREMAL SETS

IN HILBERT SPACES

Nguyen Van Khiem

Faculty of Mathematics, Hanoi National University of Education

Abstract Based on our previous result and by using the technique on α-minimal and χ-minimal sets with respect to the Kuratowski and Hausdorff measures of noncompactness, we give some new geometric characterizations of extremal sets

in Hilbert spaces

Keywords: Extremal sets, Jung’s constant, Kuratowski and Hausdorff measures of noncompactness

1 Introduction

In [1] the authors introduced the notion of extremal sets of a Banach space with respect to Jung constant Given a Banach space (X, ∥ · ∥), the Jung constant of X is defined by

J (X) = sup rX(A)

d(A) : A is a bounded subset of X with diameter d(A) > 0

 ,

where d(A) = sup{∥x − y∥ : x, y ∈ A} and rX(A) = inf

y∈A

∥y − x∥ denote the diameter and the absolute Chebyshev radius of A, respectively A point c ∈ X is called a Chebyshev center of A, if rX(A) = rc(A) = sup

y∈A

∥y − c∥

Recall that a bounded subset A of a Banach space X consisting of at least two points is extremal, if rX(A) = J (X)d(A) For given an n-dimensional Euclidean space En, the Jung’s theorem asserts that J (En) =

2(n + 1). Furthermore, a bounded subset A of En is extremal if and only if A contains all vertices of a regular n-simplex with edges of length d(A) (see [2]) For H is an infinite-dimensional Hilbert space, it is well known that J (H) = √1

2 (see [3]) Therefore, a bounded subset A

Received May 15, 2022 Revised: June, 20 2022 Accepted June 28, 2022.

Contact Nguyen Van Khiem, e-mail address: nvkhiem@hnue.edu.vn

of H is extremal if and only if rH(A) = √1

2 d(A) We now recall the main result

of [1] involving geometrically characterizing extremal sets in a Hilbert space, which is an infinite-dimensional generalization of classical Jung’s theorem

Theorem 1.1 ([1]) A bounded subset A of a Hilbert space H with d(A) = d > 0 is extremal if and only if for anyε ∈ (0, d) and positive integer m, there exists an m–simplex

∆(ε, m) with vertices in A and edges of length not less than d − ε Furthermore, for such

a subsetA, we have α(A) = d and χ(A) = rH(A), where α(A) and χ(A) denote the Kuratowski and Hausdorff measures of noncompactness of A which are defined as inf {ε > 0 : A can be covered by finitely many sets of diameter ⩽ ε} and inf {ε > 0 : A can

be covered by finitely many balls of radius⩽ ε}, respectively

In [4], Domingez-Benavides introduced the notions of α-minimal and χ-minimal sets We say that an infinite set A of a metric space X is α-minimal (resp χ-minimal) if for any infinite subset B of A one has α(B) = α(A) (resp χ(B) = χ(A)) A sequence {xn}n∈Nis said to be α-minimal (resp χ-minimal) if the set {xn}n∈Nis α-minimal (resp χ-minimal) For the properties of α-minimal and χ-minimal sets we refer the reader

to [4, 5, 6]

In this note we give three more geometric characterizations of extremal sets in Hilbert spaces

Theorem 1.2 Let A be a bounded subset of a Hilbert space H with diameter d(A) > 0 ThenA is an extremal set of H if and only if for any ε ∈ (0, d(A)), there exists an infinite simplex∆(ε, ∞) with vertices in A and edges of length not less than d(A) − ε

Theorem 1.3 Let A be a bounded subset of a Hilbert H with diameter d(A) > 0 Then A

is an extremal set ofH if and only if A contains a sequence {xn} satisfying the following properties

(i) {xn} is both α–minimal and χ–minimal;

(ii) {xn} converges weakly to the Chebyshev center of A in H;

(iii) α({xn}) = d(A) and χ({xn}) = rH(A)

Definition 1.1 We say that a sequence {xn} in a Hilbert space H is an asymptotically orthonormal sequence, if

lim

n→∞∥xn∥ = 1 and lim

m,n→∞

m̸=n

⟨xm, xn⟩ = 0,

where⟨·, ·⟩ denote the inner product of H

Theorem 1.4 Let A be a subset of the closed unit ball of a Hilbert space H with diameterd(A) = √

2 Then A is an extremal if and only if A contains an asymptotically orthonormal sequence

2 Two auxiliary lemmas

Lemma 2.1 If A is an extremal subset of a Hilbert space H with diameter d, then there exists a separable Hilbert subspaceH′ofH such that AH′ = A ∩ H′ is also an extremal subset ofH′ Furthermore,rH′(AH′) = rH(A) and d(AH′) = d

Proof From the proof of Theorem 1 in [1], it follows that for every positive integer m there exists a subset Am = {x1, x2, , xm} ⊂ A such that

∥xi− xj∥ > d − 1

m ∀ i ̸= j; i, j = 1, 2, , m

Take c arbitrarily in the convex hull coAm of Am, then there exist non-negative numbers

t1, t2, , tmsuch that

m

X

i=1

tixi = c and

m

X

i=1

ti = 1

For each j ∈ {1, 2, , m}, we have (1 − tj)



d − 1 m

2

⩽

m

X

i=1

ti∥xi− xj∥2 =

m

X

i=1

ti∥(xi− c) − (xj− c)∥2

=

m

X

i=1

ti ∥xi− c∥2+ ∥xj− c∥2− 2 ⟨xi− c , xj− c⟩

⩽ 2r2c(Am) + 2

* m

X

i=1

tixi− c , xj− c

+

= 2r2c(Am)

Hence,

(m − 1)



d − 1 m

2

=

m

X

j=1

(1 − tj)



d − 1 m

2

⩽ 2m.rc2(Am), or

rc(Am) ⩾

r

m − 1 2m



d − 1 m



By a result of Garkavi and Klee (see [7, 8]), the Chebyshev center of Amlies in the convex hull coAm of Am Since c is arbitrary in coAm, one gets

rH(Am) ⩾

r

m − 1 2m



d − 1 m



Cleary A∞ =

∞

S

m=1

Amis a countable subset of A From the estimates above we get

d(A∞) = d(A) = d,

rH(A∞) = √1

2 d = rH(A).

Now consider H′ = span A∞the closed subspace of H which is generated by A∞, and

AH′ = A ∩ H′ Clearly H′is a separable subspace of H Since A∞ ⊂ AH′ ⊂ A one gets

rH(A∞) ⩽ rH(AH′) ⩽ rH(A), d(A∞) ⩽ d(AH ′) ⩽ d(A)

Hence, rH(AH′) = rH(A) and d(AH′) = d(A), so AH′ is an extremal set in H By using the orthogonal projection of H on the closed subspace H′, it is easy to see that the Chebyshev center of AH ′ lies in H′ So rH(AH′) = rH′(AH′) Hence AH′ also is extremal

in H′ The proof of Lemma 2.1 is completed

Lemma 2.2 Let A be an α-minimal and χ-minimal subset of a Hilbert space Then χ(A) = √1

2α(A).

Proof From Lemma 3.5 of [4], let us omit it here

3 Proofs of the main results

Proof of Theorem 1.2 First, assume that A is an extremal set of a Hilbert space H with diameter d(A) = d > 0 By Lemma 2.1, we can assume that H is a separable Hilbert space By Theorem 1.1, one has α(A) = d and χ(A) = rH(A) = √1

2d From [4] (Propositions 3.2, 3.3) it follows that there exists a subset B of A which is both α-minimal and χ-minimal with χ(B) = χ(A) In view of Lemma 2.2 one obtains

1

√

2 =

χ(B) α(B) ⩾ χ(A)

α(A) =

1

√

2. Hence α(B) = α(A) = d By using Ramsey’s argument of (see [4], Lemma 3.4) there exists an infinite subset Bε ⊂ B, ∀ε ∈ (0, d), such that ∥x − y∥ >

d − ε ∀x, y ∈ Bε, x ̸= y So we can choose an infinite simplex ∆(ε, ∞) with vertices in Bεand its edges have length not less than d(A) − ε

Conversely, for given ε ∈ (0, d) if A contains an infinite simplex ∆(ε, ∞) with vertices in A and edges of length not less than d(A) − ε Consequently by Theorem 1.1,

A is an extremal set The proof of Theorem 1.2 is completed

Proof of Theorem 1.3 If A is an extremal set of a Hilbert space H, then from the proof

of Theorem 1.2 it follows that there exists a subset B ⊂ A which is both α–minimal and χ–minimal and such that χ(B) = χ(A) = rH(A), α(B) = α(A) = d(A) Let {xn} be a sequence in B, then {xn} also is both α–minimal and χ–minimal Furthermore α({xn}) = d(A) and χ({xn}) = rH(A) Since H is reflexive, we may assume that {xn}

converges weakly to a point, say c It is known that the function Φ : H → R defined by Φ(z) = lim sup

n→∞

∥xn− z∥ attains its unique minimum at c and Φ(c) = χ({xn}) = rH(A) (see [9], cf [4]) Thus c is the Chebyshev center of A Hence the sequence {xn} satisfies the conditions (i)–(iii)

Conversely if A contains a sequence {xn} satisfying the conditions (i)–(iii), then

by Lemma 2.2 one has

rH(A) d(A) =

χ({xn}) α({xn}) =

1

√

2. Hence A is an extremal set in H The proof of Theorem 1.3 is completed

Proof of Theorem 1.4 Assume that A is a subset of closed unit ball B(O, 1) of H with diameter d(A) =√

2 If A is an extremal, then rH(A) = 1 and O is a unique Chebyshev center of A By Theorem 1.3 there exists a sequence {xn} ⊂ A satisfying the properties (i)–(iii) Hence we have lim sup

n→∞

∥xn − O∥ = χ({xn}) = rH(A) By proceeding to a subsequence if necessary, one may assume that lim

n→∞∥xn∥ = rH(A) = 1 Since {xn}

is an α–minimal sequence and by Lemma 3.4 in [4], there exists a decreasing chain of subsequences of {xn}:

{xn} ⊃ {xn 1,1 , xn 1,2 , } ⊃ {xn 2,1 , xn 2,2 , } ⊃ · · · ⊃ {xnk,1 , xnk,2 , } ⊃ · · · satisfying

√

2 − 1

k ⩽ ∥xnk,i− xnk,j∥ ⩽√2 , ∀k ⩾ 1, ∀i ̸= j

Taking the diagonal sequence {zk}, defined by zk = xnk,k one sees that

√

2 − 1

k ⩽ ∥zp− zk∥ ⩽√2 , ∀p > k ⩾ 1

Since lim

k→∞∥zk∥ = 1 and ∥zp− zk∥2 = ∥zp∥2+ ∥zk∥2− 2 ⟨zp, zk⟩, one gets

lim

p,k→∞

p̸=k

⟨zp, zk⟩ = 0,

i.e {zk} is an asymptotically orthornormal sequence

Conversely, if d(A) =√

2 and A contains an asymptotically orthonormal sequence {zk}, then we have

lim

p,k→∞

p̸=k

∥zp− zk∥ =√2 = d(A)

Hence for every ε ∈ (0, d(A)) there exists a positive integer n0such that

∥zp− zk∥ > d(A) − ε, ∀p > k ⩾ n0

It follows that the infinite simplex ∆(ε, ∞) with vertices zn 0, zn0+1, zn0+2, has all edges of length not less than d(A) − ε Therefore, by Theorem 1.1, A is an extremal set The proof of Theorem 1.4 is completed

4 Conclusions

In this paper, we study the geometrical properties of extremal sets in Hilbert spaces Based on our results in [1], and by using the technique on α-minimal and χ-minimal sets with respect to the Kuratowski and Hausdorff measures of noncompactness, we obtain three new geometric characterizations of extremal sets in Hilbert spaces

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